The eigenvalues and eigenvectors of the above matrix (REF ) can be determined analytically [1]} by solving \((L_4 - \lambda _i I) \mathbf {v}_i = 0 ~~\forall ~i \in [1,n]\) , where \(\lambda _i\)  is the \(i\) th eigenvalue and \(\mathbf {v}_i\)  is the corresponding eigenvector. The eigenvalues of Laplacian \(L_4\)  are the roots of its characteristic polynomial, which is given as follows.
\(P_{L_4}(\lambda ) = \lambda (\lambda - 4) (\lambda ^2 - 4 \lambda - 2 \alpha \lambda - 2 \beta \lambda + 3 + 5 \alpha + 5 \beta + 3 \alpha \beta )=0\) 
